How to randomise vertex position along an axis
You can use Proportional Editing set to a random falloff.

Basically, select the vertices you want affected, press Ctrl + I, to invert the selection, and then H to hide those vertices. Now, select a vertex of the ones left, press O, to turn Proportional Editing on, and, in the 3D view header, set the falloff to random:



Then, press G, to move, followed by one of X, Y or Z, either once or twice, to limit to the axis you want. Now, use the scroll wheel to enlarge the affected area so that all the vertices are affected and move the mouse until you get the results you want.
Then, press  Alt + H to show all hidden vertices again.